The ANF of the Composition of Addition and Multiplication mod 2n with a Boolean Function

  • An Braeken
  • Igor Semaev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3557)


Compact formulas are derived to represent the Algebraic Normal Form (ANF) of \(f(\bar{x} + \bar{a}~mod~2^{n})\) and \(f(\bar{x} \times \bar{a}~mod~2^{n})\) from the ANF of f, where f is a Boolean function on \(\mathbb{F}^{n}_{2}\) and \(\bar{a}\) is a constant of \(\mathbb{F}^{n}_{2}\). We compare the algebraic degree of the composed functions with the algebraic degree of the original function f. As an application, the formula for addition modulo 2 n is applied in an algebraic attack on the summation generator and the E 0 encryption scheme in the Bluetooth keystream generator.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • An Braeken
    • 1
  • Igor Semaev
    • 2
  1. 1.Department Electrical Engineering, ESAT/COSICKatholieke Universiteit LeuvenHeverlee-LeuvenBelgium
  2. 2.Selmer Center, Inst. for InformatikkUniversity of BergenBergenNorway

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